reserve M for non empty MetrSpace,
        F,G for open Subset-Family of TopSpaceMetr M;
reserve L for Lebesgue_number of F;
reserve n,k for Nat,
        r for Real,
        X for set,
        M for Reflexive non empty MetrStruct,
        A for Subset of M,
        K for SimplicialComplexStr;

theorem
  for S be finite non empty Subset of M
    ex p,q be Point of M st p in S & q in S & dist(p,q) = diameter S
 proof
  let S be finite non empty Subset of M;
  set q=the Element of S;
  reconsider q as Point of M;
  deffunc D(Point of M,Point of M)=dist($1,$2);
  set X={D(x,y) where x is Point of M,y is Point of M:x in S & y in S};
  A1: now let x be object;
   assume x in X;
   then ex y be Point of M,z be Point of M st x=D(y,z) & y in S & z in S;
   hence x is real;
  end;
  A2: D(q,q) in X;
  A3: S is finite;
  X is finite from FRAENKEL:sch 22(A3,A3);
  then reconsider X as real-membered non empty finite set by A1,A2,
MEMBERED:def 3;
  reconsider sX=sup X as Real;
  sX in X by XXREAL_2:def 6;
  then consider p,q be Point of M such that
   A4: sX=D(p,q) & p in S & q in S;
  now let x,y be Point of M;
   assume x in S & y in S;
   then D(x,y) in X;
   hence D(x,y)<=sX by XXREAL_2:4;
  end;
  then A5: diameter S<=sX by TBSP_1:def 8;
  sX<=diameter S by A4,TBSP_1:def 8;
  hence thesis by A4,A5,XXREAL_0:1;
 end;
